I'm very surprised noone's yet listed Sheldon Axler's Linear Algebra Done Right - unlike Strang and Lang, which are really great books, Linear Algebra Done Right has a lot of "common sense", and is great for someone who wants to understand what the point of it all is, as it carefully reorders the standard curriculum a bit to help someone understand what it's all about.
With a lot of the standard curriculum, you can get stuck in proofs and eigenvalues and kernels, before you ever appreciate the intuition and applications of what it's all about. This is great if you're a typical pure math type who deals with abstraction easily, but given the asker's description, I don't think that a rigorous pure math course is what he/she's asking for.
For the very practical view, yet also not at all sacrificing depth, I don't think you can do better than Linear Algebra Done Right - and if you are thirsty for more, after you've tried it, Lang and Strang are both great texts.